Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.1.1.9 (Full Subcategories of Nonunital Monoidal Categories). Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $(\otimes , \alpha )$, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}_0$, the tensor product $X \otimes Y$ also belongs to $\operatorname{\mathcal{C}}_0$. Then $\operatorname{\mathcal{C}}_0$ inherits a nonunital monoidal structure, with tensor product functor given by the composition

\[ \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\xrightarrow {\otimes } \operatorname{\mathcal{C}} \]

(which factors through $\operatorname{\mathcal{C}}_0$ by hypothesis), and associativity constraints given by those of $\operatorname{\mathcal{C}}$.